On weighted means and their inequalities

In (Pal et al. in Linear Multilinear Algebra 64(12):2463–2473, 2016), Pal et al. introduced some weighted means and gave some related inequalities by using an approach for operator monotone functions. This paper discusses the construction of these weighted means in a simple and nice setting that immediately leads to the inequalities established there. The related operator version is here immediately deduced as well. According to our constructions of the means, we study all cases of the weighted means from three weighted arithmetic/geometric/harmonic means by the use of the concept such as stable and stabilizable means. Finally, the power symmetric means are studied and new weighted power means are given.


Introduction
The mean inequalities arise in various contexts and attract many mathematicians by their developments and applications. It has been proved throughout the literature that the mean theory is useful in a theoretical point of view as well as in practical purposes.

Standard weighted means
As usual, we understand by (binary) mean a map m between two positive numbers such that min(a, b) ≤ m(a, b) ≤ max(a, b) for any a, b > 0. Continuous (symmetric/homogeneous) means are defined in the habitual way. If m is a mean, then we define its dual by m * (a, b) = (m(a -1 , b -1 )) -1 . It is easy to see that if m is continuous,(resp. symmetric, homo- It is obvious that (iii) implies (ii). The mean m := m 1/2 is called the associated symmetric mean of m v . It is not hard to check that if m v is a weighted mean then so is m * v . The standard weighted means are recalled in the following. The weighted arithmetic mean For v = 1/2, they coincide with a∇b, a b, and a!b, respectively. These weighted means satisfy for any a, b > 0 and v ∈ [0, 1]. These weighted means are all strict provided v ∈ (0, 1).

Two weighted means
Recently, Pal et al. [6] introduced a class of operator monotone functions from which they deduced other weighted means, namely the weighted logarithmic mean defined by and the weighted identric mean given by (1.4) and hold for any a, b > 0 and v ∈ [0, 1]. Otherwise, Furuichi and Minculete [3] gave a systematic study from which they obtained a lot of mean inequalities involving L v (a, b) and I v (a, b). Some of their inequalities are refinements and reverses of (1.5) and (1.6).
The outline of this paper is organized as follows: In Sect. 2 we give simple forms for L v (a, b) and I v (a, b), and mean inequalities are obtained in a fast and nice way. We also deduce two other weighted means from L v (a, b) and I v (a, b). Section 3 is devoted to investigating a general approach in service of weighted means. We then obtain more weighted means in another point of view. Section 4 displays the operator version of the previous weighted means as well as their related inequalities. In Sect. 5 we recall the standard power means known in the literature, and we use, in Sect. 6, our approach for obtaining some new weighted means associated with the previous power means.

Another point of view for defining L v (a, b) and I v (a, b)
We preserve the same notations as in the previous section. The expressions (1.3) and specially (1.4) seem to be hard in computational context. We will see that we can rewrite them in other forms having convex characters.

Simple forms of L v (a, b) and I v (a, b)
The key idea of this section turns out to be the following result.
Proof Starting from the middle expression of (2.1) and using the definition of L(a, b) and a v b, we get the desired result after simple algebraic manipulations as follows: In a similar way we get (2.2) as In what follows we will see that inequalities (1.5) and (1.6) can be immediately deduced from (2.1) and (2.2), respectively. In fact we will prove more.
The two right inequalities of (2.3) are those of (1.5). We will prove them again by using (2.1). Indeed, (2.1) with the help of (1.1), and then (1.2) yields We now prove the two left inequalities of (2.3). Again, (2.1) with (1.1) and then (1.2) implies that We left to the reader the routine task to prove (2.4) in a similar manner. In order to emphasize even more the importance of (2.1) and (2.2), we present below more results. These results investigate some inequalities refining the right inequalities in (2.3) and (2.4). We need the following lemma. Proof It is a simple exercise of real analysis.

Theorem 2.5 For any a, b > 0 and v
Proof We prove the first inequality in (2.5). Since the map The second and third inequalities of (2.5) follow from (1.1).
To prove the second inequality of (2.6), we write by using the previous lemma 2), yields the second inequality of (2.6). To prove the first inequality of (2.6), we write after a simple computation. The proof is finished.

Two other weighted means
A natural question arises from the previous subsection: do we have a weighted mean We can also put the following question: do we have a weighted mean P v (a, b) such that In what follows we will answer the two preceding questions. Recall that L * denotes the dual of the logarithmic mean L, and L * v is the dual of the weighted logarithmic mean L v , as previously defined. Similar sentence for I * and I * v . We will establish the following result.
Proof We can of course assume that v ∈ (0, 1). If in (2.1) we replace a and b with a -1 and b -1 , respectively, then we get Taking the inverses side by side and using the definition of the weighted harmonic mean, we infer that Now, let us set If v ∈ (0, 1) is fixed, for any a > 0 and x > 0, it is easy to see that there exists a unique b > 0 such that a v b = x. This means that M is well defined by (2.12). Further, if we remark that It follows that M is the dual of the logarithmic mean L. Following (2.11) and (2.7), the associated weighted mean M v of M is such that ) is the dual of the weighted logarithmic mean L v . We left to the reader the task to prove (2.10) in a similar manner.
Remark 2.7 After this, let us observe the following question: is L v the unique weighted mean satisfying (2.1)? In the next section, we answer this question via a general point of view. A similar question can be put for (2.2), (2.9), and (2.10).

Weighted means in a general point of view
As already pointed before, we investigate here a study that shows how to construct some weighted means in a general point of view. Indeed, p v (a, b) and q v (a, b) are given. Once the (symmetric) mean M is found, we obtain

Position of the problem
Our aim here is to answer the previous question in its general form. We need to recall some notions and results as background material that we will summarize in the next subsection.

Stable and stabilizable means
We recall here in short the concept of stable and stabilizable means introduced in [7,8,10]. Let m 1 , m 2 , and m 3 be three given symmetric means. For all a, b > 0, the resultant mean map of m 1 , m 2 , and m 3 is defined by [7] R (m 1 , m 2 , m 3 )(a, b) = m 1 m 2 a, m 3 (a, b) , m 2 m 3 (a, b), b .
A symmetric mean m is called stable if R(m, m, m) = m and stabilizable if there exist two nontrivial stable means m 1 and m 2 such that R(m 1 , m, m 2 ) = m. We then say that m is (m 1 , m 2 )-stabilizable. If m is stable, then so is m * , and if m is (m 1 , m 2 )-stabilizable, then m * is (m * 1 , m * 2 )-stabilizable. The tensor product of m 1 and m 2 is the map, denoted m 1 ⊗ m 2 , defined by m 2 (a, b, c, d) = m 1 m 2 (a, b), m 2 (c, d) .
A symmetric mean m is called cross mean if the map m ⊗2 := m ⊗ m is symmetric in its four variables. Every cross mean is stable, see [7], and the converse is still an open problem.
It is worth mentioning that the operator version of the previous concepts as well as their related results has been investigated in a detailed manner in [11]. It has been proved there that every cross operator mean is stable but the converse does not in general hold provided that the Hilbert operator space is of dimension greater than 2.

Theorem 3.2
Let m 1 and m 2 be two symmetric means such that m 1 ≤ m 2 (resp. m 2 ≤ m 1 ). Assume that m 1 is a strict cross mean. Then there exists one and only one (m 1 , m 2 )stabilizable mean m such that m 1 ≤ m ≤ m 2 (resp. m 2 ≤ m ≤ m 1 ).

The main result
Now, we are in the position to answer our previous question as recited in the following result. This means that M is (q, p)-stabilizable. According to Theorem 3.2, such M exists and is unique. Since p v and q v are given, we then deduce the existence and uniqueness of M v satisfying (3.1). The proof is finished.
Following Theorem 3.1, the symmetric means a∇b, a b, a!b are cross means, and so they are stable. From the preceding theorem we immediately deduce the following corollary.
then we have the same conclusion as in the previous theorem.
The following examples discuss these cases in detail.
then M = I * and M v is given by (2.10).

Example 3.6 For the three cases
. In a similar way as previously, we show that the associated mean M is here given by M = L * the dual logarithmic mean. The associated weighted mean M v is defined by Also, from this latter relation we can verify that The previous examples are summarized in Table 1.

Operator version
The operator version of the previous weighted means as well as their related operator inequalities have been also discussed in [6]. By using their approach for operator monotone functions and referring to the Kubo-Ando theory [5], they studied the analogues of Following the Kubo-Ando theory [5], there exists a unique one-to-one correspondence between operator means and operator monotone functions. More precisely, an operator mean m in the Kubo-Ando sense is such that are known in the literature as the weighted arithmetic mean, the weighted geometric mean, and the weighted harmonic mean of A and B, respectively. If v = 1/2, they are simply denoted by A∇B, A B, and A!B, respectively. The previous operator means satisfy the following double inequality: (4. 2) The weighted logarithmic mean and the weighted identric mean of A and B can be, respectively, defined through:  Since all the involved operators in (4.6) and (4.7) are operator means in the sense of (4.1), then by Theorem 2.2 we immediately deduce the following result as well.
By the same arguments as previous, the operator version of Theorem 2.5 is immediately given in the following statement.

Power symmetric means
This section deals with some weighted means for power symmetric means in one or two parameters. Let a, b > 0 and p, q be two real numbers. We recall the following: • The power binomial mean defined by All the previous power means are symmetric in a and b. Also, remark that S p,q is symmetric in p and q. Otherwise, the power binomial mean B p is stable for any p ∈ R and the following result holds, see [9].
Theorem 5.1 For any p, q ∈ R, the Stolarsky mean S p,q is (B q-p , B p )-stabilizable.
The previous theorem, when combined with (5.7) and a simple argument of continuity, immediately implies the following, see also [7].

Corollary 5.2
For all real number p, the following assertions hold: (i) The power mean L p is (B p , ∇)-stabilizable, while D p is (∇, B p )-stabilizable.
(ii) The power mean I p is ( , B p )-stabilizable, while L p is (B p , )-stabilizable. Now, let us observe the following remark which is of interest.
Remark 5.3 Since S p,q = S q,p , we can also say that S p,q is (B p-q , B q )-stabilizable. This, with (5.7), implies also that L p is (B -p , B p+1 )-stabilizable, D p is (!, B p+1 )-stabilizable, L p is (B -p , B p )-stabilizable, and no news for I p . Obviously, (i) and (ii) of Corollary 5.2 are simpler than these latter statements.

Some new weighted power means
In this section we investigate the weighted means of the previous power means. The weighted power binomial mean can be immediately given by This, with the results presented in the preceding section, will allow us to construct some new weighted power means. Recall that m v is called weighted mean if it satisfies the conditions: m v is a mean for any v ∈ [0, 1], m := m 1/2 is a symmetric mean, and m v (a, b) = m 1-v (b, a) for any a, b > 0 and v ∈ [0, 1]. We then say that m v is a m-weighted mean and m is the symmetric mean of m v . It is obvious that for any weighted mean m v its associated